Above I thought that the pedal extenders worked for me -- reducing knee soreness -- because they encouraged me to ride rotating my toes outwards.
I have now come to the conclusion that the pedal extenders rotated my toes outwards, but it was fighting against this outward rotation, and rotating inwards, preventing myself from going bow legged, that prevented sore knees. In other words, pedal extenders encouraged internal rotation of the femur, to keep my legs and pedaling straight.
The extenders worked for a while but my femural rotational strength got worse.
I forgot about pedal extenders and tried all sorts of exercises squats, one leg Romanian squats (on a wobbly board and on a rotating platform), CARs, the McGill “big three” (hip abductions, flying dogs, side planks), hydrants, donkey and ninja kicks, and their combinations by the thousand.
Eventually, remembering the way that pedal extenders made me rotate my legs, feeling that scything, golf swings, and The Twist (Chubby Checker) helped, I concentrated on rotational strength, and with a
banded leg twisting exercise, I have found that my legs are feeling a lot better - no sore knees, no hip problems, a real sense of leg youth and stability.